# LMIs in Control/pages/Full-State Feedback Optimal Control Hinf LMI

**Full State Feedback Optimal Control**Edit

Full State Feedback is a control technique that attempts to place the system's closed loop system poles in specified locations based off of performance specifications given. methods formulate this task as an optimization problem and attempt to minimize the norm of the system. In a single-input single-output (SISO) system this norm represents the maximum gain on a magnitude Bode plot. In the case of multi-input multi-output (MIMO) systems it can be interpreted as maximum response to a perturbation introduced to the system. In either, by minimizing the we are minimizing the worst case effect of a disturbance to the system, whether it is noise or another perturbation.

**The System**Edit

The system is represented using the 9-matrix notation shown below.

where is the state, is the regulated output, is the sensed output, is the exogenous input, and is the actuator input, at any .

The lower linear fractional transformation (LFT) is used to implement a controller into the system. The lower LFT is denoted as and is formed by with . For full-state feedback we consider a controller of the form . This is a special case where and results in a controller of the form .

**The Data**Edit

, , , , , , , , are known.

**The LMI:**Full State Feedback Optimal Control LMIEdit

The following are equivalent.

1) There exists a such that

2) There exists and such that

- .

Then .

**Conclusion:**Edit

The above LMI, if feasible, will determine the bound on the norm of the system. In addition to this is also determined allowing the closed loop system to be determined using the controller found during the optimization.

**Implementation**Edit

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_Hinf.m

**Related LMIs**Edit

**External Links**Edit

- LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
- LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
- LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
- LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
- A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Chapter 7.